The opera Impresario, who's very sensitive about his operas, does this when an act goes wrong or when it may go wrong. It is probable, though, that his "face palm" is more of a corny, theatrical and/or dramatical gesture of distress, (you know, that old cover-your-face-with-your-arm/hand-whilds-wistfully-going-"Oh" thingy,) rather than the common, slightly more humorous usage of the gesture. This would make sense seeing as he is, well, the impresario of an opera.
A straight example of this occurs when Gau leads Sabin and Cyan to a cave, to show them a "shiny shiny". Once they arrive... he forgets where it is, cuing a facepalm.
Lulu responds to a lot of Tidus's and Wakka's shenanigans by burying her face in her hand.
When Yuna emerges from an Al Bhed ship, having beaten up its crew - rendering your party's rescue pointless - Lulu, Kimahri and even Tidus are shown doing this at once. Then the normal victory music plays.
Fire Emblem: Kent, whenever Sain starts making his moves on a girl. See on the right hand side here◊.
In Dead Space, the main character does a facepalm after he learns that his girlfriend, who he thought was alive, was Dead All Along. A sad facepalm indeed.
In the Professor Layton series, both Layton and Luke do this if they get a puzzle wrong. Layton pulls the brim of his tophat over his eyes, while Luke does the classic forehead slap.
In Utawarerumono This is Eruruw's default expression for when Hakuoro does something stupid — but not stupid enough to get seriously angry over (which he does often).
World of Warcraft added a /facepalm emote in patch 3.0.2, although it only produces a predefined text message and not animated. It's also one of the few emotes the game's statistics page will track on your character; they expect you to use it that much.
Anarchy Online as well added a similar command, and animated it.
Guild Wars also has a facepalm emote, though the command for it is /doh.
In 2011, players who liked the City of Heroes Facebook page on July 21 or between August 17-31 received the /facepalm emote ingame.
The Spy class on a losing team can be seen face palming if he is still alive during the aptly named "Humiliation" phase. He also does this in one of the official comics: In "A Smissmas Story", he face palms and rests his head on the desk when Soldier (acting as Spy and Scout's lawyer) encourages Scout to go with a completely ludicrous testimony.
The Sniper does the hat version in his "Meet the Sniper" video when he's trying to talk to his father on the phone.
In BlazBlue Jin and Hakumen facepalm if they lose to Time Up.
In another Arc System Works example, this is Baiken's taunt in Guilty Gear.
Dark Forces Saga: Kyle Katarn will do this when he gets bored game-over due to some reason other than his own death.
In Sonic Colors (Wii version), Sonic does a facepalm in the cutscene where Tails is attempting to figure out what the Wisps are saying. ("He says his name is Talks-a-lot and he's from a far off soda...") Yacker (the Wisp Tails was trying to translate for) also facepalms. At the exact same time as Sonic does.
Sonic facepalms up to twice more throughout Sonic Generations. First time is if Chemical Plant Act 2 is the second act of that level to be completed—after Amy speaks her line, Modern Sonic facepalms. He facepalms once more during the Reveal.
If you're watching him, Raz pulls this off during the boss fight in Black Velvetopia. It's not the main focus, but not easy to miss, either — just pay attention to him while Edgar reverts to El Odio.
While we're on the topic, Sasha gets the chance at the end of the game, when Ford's psitanium wears off and Chef Cruller returns.
Also done in Brütal Legend (which is from the creators of Psychonauts). During the intro, the band Eddie was a roadie for starts their new song with a really badass and edgy guitar riff...only to transition into sissy pop singing. Eddie then facepalms.
Jansen the drunken miner: My farm supplied eggplant to half of South Reach! Other drunken miner: Oh yeah? Well, my eggplant supplied half the bastards in South Reach! (drunken laughter) Hawke:(facepalm)
Sarcastic Hawke pulls another one during the quest where he/she looks for an escaped prisoner in Act 1.
Lia the kidnappee: He didn't mean to hurt me! There are demons who made him do all these horrible things! Hawke:(facepalm) I should remember this one. "A demon made me do it."
Ken does this in Super Puzzle Fighter II Turbo when his playfield his halfway full of gems.
In Star Wars: The Old Republic, on one of the Sith Warrior missions you have the option of sleeping with Lord Grathan's wife. Later you can brag about it to Darth Baras, prompting a facepalm from Vette.